Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Volumes and Integrals over Polytopes Jes´ us A. De Loera, UC Davis

July 16, 2009

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Why compute the volume and its cousins?

Meet Volume The (Euclidean) volume V (R) of a region of space R is real non-negative number defined via the Riemann integral over the regions.

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Why compute the volume and its cousins?

Meet Volume’s Cousins In the case when P is an n-dimensional lattice polytope (i.e., all vertices have integer coordinates) we can naturally define a normalized volume of P, NV (P) to be n!V (P). EXAMPLE: P = {(x, y ) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} NV (P) = 2! · 1 = 2. Given polytopes P1 , . . . , Pk ⊂ Rn and real numbers t1 , . . . , tk ≥ 0 the Minkowski sum is the polytope t1 P1 + · · · + tk Pk := {t1 v1 + · · · + tk vk : vi ∈ Pi } .

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Why compute the volume and its cousins?

EXAMPLE

Theorem(H. Minkowski) There exist MV (P1a1 , . . . , Pkak ) > 0 (the mixed volumes) such that V P(t1 P1 + · · · + tknPk
) = ak a1 a2 ak a1 a1 +···+ak =n a1 ,...,ak MV (P1 , . . . , Pk )t1 t2 · · · tk .

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Why compute the volume and its cousins?

A few reasons to compute them (for algebraic geometers) If P is an integral polytope, then the normalized volume of P is the degree of the toric variety associated to P. (for computational algebraic geometers) Let f1 , . . . , fn be polynomials in C[x1 , . . . , xn ]. Let New (fj ) denote the Newton polytope of fj , If f1 , . . . , fn are generic, then the number of solutions of the polynomial system of equations f1 = 0, . . . , fn = 0 with no xi = 0 is equal to the normalized mixed volume n!MV (New (f1 ), . . . , New (fn )). (for Combinatorialists ) Volumes count things! P P CRm = {(aij ) : i aij = 1, j aij = 1, with aij ≥ 0 but aij = 0 when j > i + 1 }, then NV (CRm ) = product of first (m − 2) Catalan numbers. (D. Zeilberger).

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Computational Complexity of Volume

Do we need limits to define volumes of polytopes?

1 volume of egyptian pyramid = (area of base) × height 3

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Computational Complexity of Volume

Easy and pretty in some cases...

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Computational Complexity of Volume

In general, proofs seem to rely on an infinite process!

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Computational Complexity of Volume

But not necessary in dimension two!

Polygons of the same area are equidecomposable, i.e., one can be partitioned into pieces that can be reassembled into the other.

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Computational Complexity of Volume

Hilbert’s Third Problem Are any two convex 3-dimensional polytopes of the same volume equidecomposable?

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Computational Complexity of Volume

NOT always!!! We need calculus to define the volume of

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Computational Complexity of Volume

high-dimensional polytopes. It is hard to compute the volume of a vertex presented polytopes (Dyer and Frieze 1988, Khachiyan 1989). Number of digits necessary to write the volume of a rational polytope P cannot always be bounded by a polynomial on the input size. (J. Lawrence 1991). Theorem (Brightwell and Winkler 1992) It is #P-hard to compute the volume of a d-dimensional polytope P represented by its facets. We even know that it is hard to compute the volume of zonotopes (Dyer, Gritzmann 1998). Thus computing mixed volumes, even for Minkowski sums of line segments, is already hard! For convex bodies, deterministic approximation is already hard, but randomized approximation can be done efficiently (work by Barany, Dyer, Elekes, Furedi, Frieze, Kannan,

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Via Triangulations Via Rational Functions for Lattice Points

simplices

SIMPLICES are d-dimensional polytopes with d + 1 vertices. E.g., triangles, tetrahedra, etc. The volume of a (Euclidean) simplex is given by a fast determinant calculation. To compute the volume of a polytope: divide it as a disjoint union of simplices, calculate volume for each simplex and then add them up!

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Via Triangulations Via Rational Functions for Lattice Points

Triangulations: Enough to know how to do it for simplices!

Theorem: For all polytopes in fixed dimension d their whole volume can be computed in polynomial time.

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Via Triangulations Via Rational Functions for Lattice Points

The size of a triangulation changes! Triangulations of a convex polyhedron come in different sizes! i.e. the number of simplices changes.

6

8

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Via Triangulations Via Rational Functions for Lattice Points

Counting lattice points to approximate volume Lattice points are those points with integer coordinates: Zn = {(x1 , x2 , . . . , xn )|xi integer} We wish to count how many lie inside a given polytope! Let P be a convex polytope in Rd . For each integer n ≥ 1, let

nP = {nq|q ∈ P}

P

3P

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Via Triangulations Via Rational Functions for Lattice Points

For P a d-polytope, let

i(P, n) = #(nP ∩ Zd ) = #{q ∈ P |nq ∈ Zd } This is the number of lattice points in the dilation nP.

Volume of P = limitn→∞

i(P, n) nd

At each dilation we can approximate the volume by placing a small unit cube centered at each lattice point:

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Via Triangulations Via Rational Functions for Lattice Points

Lawrence’s Style Volume Formulas Theorem (J. Lawrence 1991) Let P be a simple d-polytope given by {x ∈ Rd : bi − ait x ≥ 0, i = 1 . . . m}. Suppose that c is a vector such that the dot produt of c with any edge of P is non-zero. Then the volume of P equals vol(P) =

(hc, v i)d 1 X d! δ v γ1 γ2 · · · γd v ∈V (P)

where if indices of the constraints that are binding at v are i1 , . . . , id then γi ’s are such c = γ1 ai1 + γ2 ai2 + · · · + γn aid and δv = |det([ai1 , ai2 , . . . , aid ])|.

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

From volume to Integration Still people need to compute integrals exactly!!!

Integration of polynomials: Given P be a d-dimensional rational polytope inside Rn and let f ∈ Q[x1 , . . . , xn ] be a polynomial with rational coefficients.

Compute the EXACT value of the integral

R P

f dm?

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

From volume to Integration Still people need to compute integrals exactly!!!

Example

17 111 13 If we integrate the monomial R x 17y 111z 13over the three-dimensional standard simplex ∆. Then ∆ x y z dxdydz equals exactly

1 317666399137306017655882907073489948282706281567360000

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

From volume to Integration Still people need to compute integrals exactly!!!

Why exact integration? Integrals over polytopes arise in probability, statistics, algebraic geometry, combinatorics, symplectic geometry. Already computing volumes is a very important subroutine. Despite the success of APPROXIMATE integration, still EXACT integration is necessary. Example: Computation of marginal likelihood integrals in model selection. Example: Statisticians used BIC, Laplace, Montecarlo approximations in concrete 6 variable problems. They say “Problems are too hard for exact methods”. approximation leads to model answer. My point: Exact integration useful for calibration!!!

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

From volume to Integration Still people need to compute integrals exactly!!!

TECHNICAL DETAILS...

The input simplex ∆: encoding of ∆ is given by the number of the dimension d, and the largest binary encoding size of the coordinates among vertices. For simplicity assume the polytope P is full dimension n, in Rn dm is the standard Lebesgue measure, which gives volume 1 to the fundamental domain of the lattice Zn . For this dm, every integral of a polynomial function with rational coefficients will be a rational number.

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

From volume to Integration Still people need to compute integrals exactly!!!

How to represent a polynomial in a computer? The input polynomial: requires that one specifies concrete data structures for reading the input polynomial and to carry on the calculations. Three main possibilities: dense representation: polynomials are given by a list of the coefficients of all monomials up to a given total degree M. sparse representation: Polynomials are specified by a list of exponent vectors of monomials with non-zero coefficients, together with their coefficients. straight-line program Φ if polynomial is a finite sequence of polynomial functions of Q[x1 , . . . , xn ], namely q1 , . . . , qk , such that each qi is either a variable x1 , . . . , xn , an element of Q, or either the sum or the product of two preceding polynomials in the sequence and such that qk = f .

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Good News Bad News

Best News: Fast Integration for powers of linear forms

Theorem: There exists a polynomial-time algorithm for the following problem. Input: numbers d, M ∈ N. affinely independent rational vectors s1 , . . . , sd+1 ∈ Qd in binary encoding, a power of a linear form h`, xiM R Output:, in binary ∆ h`, xiM dm.

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Good News Bad News

From fixed number of linear forms to fixed degree.

We can also deal with arbitrary polynomials of fixed degree. Write a polynomial as a sum of powers of linear forms. Explicit formula with at most 2M terms. md m2 x1m1 xP 2 · · · xd =

1 |m|−|p| m1 · · · md (p x + · · · + p x )|m| . (−1) 1 1 d d 0≤p ≤m p1 pd |m|! i i

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Good News Bad News

Integration of arbitrary powers of quadratic forms is NP-hard The clique problem (does G contain a clique of size ≥ n) is NP-complete. (Karp 1972). Theorem [Motzkin-Straus 1965] G a graph with P clique number ω(G ). QG (x) := 21 (i,j)∈E (G ) xi xj . Function on standard simplex in R|V (G )| . 1 Then kQG k∞ = 12 (1 − ω(G ) ). Lemma Let G a graph with d vertices. For p −1)d 3 ln(32d 2 ), the clique number ω(G ) is equal to  ≥ 4(e 1 p 1−2kQG kp . (L -norm, Holder inequality).

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Valuations A function S on polyhedra is a valuation. If it is a linear map from the vector space of characteristic functions χ(pi ) of any polyhedra into a field. P Thus if polyhedra pi satisfy a linear relation i ri χ(pi ) = 0, then X ri S(pi ) = 0, i

Example: χ(p1 ∪ p2 ) + χ(p1 ∩ p2 ) − χ(p1 ) − χ(p2 ) = 0,

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Two important valuations for polyhedra p (convex) polyhedron, rational (lattice Λ). X S(p)(ξ) := e hξ,xi x∈p∩Λ

generating function for lattice points of p. Z I (p)(ξ) := e hξ,xi dm. p

when integral and series converge. If p contains a line, then S(p) := 0 and I (p) := 0. IMPORTANT FACT: When p is a simplicial cone easy to write.

Volumes of Polytopes: FAMILIAR AND USEFUL Volume of Polytopes: NOT AS EASY AS THEY MAY SEEM! But, How to compute the volumes anyway? How to Integrate a Polynomial over a Convex Polytope New Techniques for Integration over a Simplex Ideas to integrate fast and more

Sums S(p) in dim 1 For the real line we have X

e nξ +

n>s

X

e nξ =

n